Analysis of weak solution of Euler-Bernoulli beam with axial force

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• 作者：Bidisha Kundu ^{bidisha@aero.iisc.ernet.in} ; Ranjan Ganguli ; ^{ganguli@aero.iisc.ernet.in} ; ^{drganguli@gmail.com}
• 关键词：Rotating Euler&ndash ; Bernoulli beam ; Weak formulation ; Cantilever beam ; Galerkin method ; Vibration ; Buckling
• 刊名：Applied Mathematics and Computation
• 出版年：2017
• 出版时间：1 April 2017
• 年：2017
• 卷：298
• 期：Complete
• 页码：247-260
• 全文大小：799 K
• 卷排序：298

文摘

In this paper, we discuss about the existence and uniqueness of the weak form of the non-uniform cantilever Euler–Bernoulli beam equation with variable axial (tensile and compressive) force. We investigate the reason of the buckling from the coercivity analysis. The frequencies of the beam with tensile force are found by the Galerkin method in the Sobolev space H2 with proper norm. Using this method, a system of ordinary differential equations in time variable is formed and the corresponding mass and stiffness matrices are constructed. A very general form of these matrices, which is very simple and suitable for calculations, is derived here with a standard basis. Numerical results for rotating beams with polynomial stiffness and mass variation, typical of wind turbine and helicopter rotor blades, are obtained. These results match well with the published literature. A new polynomial generating set is found. Using two elements of this set, a formula to find the eigenfrequencies is derived. The proposed approach is easy to implement in symbolic computing software.